Rotating-Wave Approximation

The rotating-wave approximation keeps near-resonant terms in a driven quantum system and drops terms that rotate rapidly in the relevant interaction or rotating frame. The exact frame change is developed in Rotating Frames; this page owns the additional approximation. It turns a time-dependent Hamiltonian into a simpler effective Hamiltonian when the drive is weak, near resonance, and observed over times long compared with the fast oscillation period. For a first encounter with the resulting two-level population motion, see Rabi Oscillations: First Encounter.

For an atomic implementation, Two-Level Atom owns the separate projection from a multilevel spectrum and the laboratory Rabi-frequency, phase, and detuning conventions used before this approximation is tested.

The Time-Dependent Two-Level Systems Notebook provides an executable laboratory-frame comparison that separates time-step convergence from the retained pointwise RWA discrepancy.

The approximation is common in atomic physics, nuclear magnetic resonance, quantum optics, and quantum information. It is powerful, but it is not automatic: counter-rotating terms can produce measurable shifts and can dominate outside the weak near-resonant regime.

Driven Two-Level System

Consider a two-level system with

$$H_0 = \frac{\hbar\omega_0}{2}\sigma_z$$

and a linearly oscillating drive

$$V(t) = \hbar\Omega\cos(\omega t)\sigma_x.$$

Using

$$\sigma_x=\sigma_++\sigma_-, \qquad \cos(\omega t) = \frac12 \left(e^{i\omega t}+e^{-i\omega t}\right),$$

the interaction-picture perturbation is

$$\begin{aligned} V_I(t) = \frac{\hbar\Omega}{2} \big[ &\sigma_+e^{i(\omega_0+\omega)t} +\sigma_+e^{i(\omega_0-\omega)t} \\ &+\sigma_-e^{-i(\omega_0-\omega)t} +\sigma_-e^{-i(\omega_0+\omega)t} \big]. \end{aligned}$$

Near resonance,

$$\Delta=\omega_0-\omega$$

is small compared with $\omega_0+\omega$. The slowly rotating terms are

$$\sigma_+e^{i\Delta t}, \qquad \sigma_-e^{-i\Delta t}.$$

The counter-rotating terms oscillate with angular frequency approximately $2\omega_0$ and average to small corrections when the drive is weak.

RWA Hamiltonian

Dropping the fast terms gives

$$V_I^{\mathrm{RWA}}(t) = \frac{\hbar\Omega}{2} \left( \sigma_+e^{i\Delta t} +\sigma_-e^{-i\Delta t} \right).$$

Equivalently, in a frame rotating at the drive frequency, the effective Hamiltonian can be written as

$$H_{\mathrm{RWA}} = \frac{\hbar\Delta}{2}\sigma_z +\frac{\hbar\Omega}{2}\sigma_x,$$

up to an overall phase convention for the rotating frame. This Hamiltonian generates Rabi oscillations with generalized frequency

$$\Omega_R = \sqrt{\Omega^2+\Delta^2}.$$

On exact resonance, $\Delta=0$, the population oscillates at the drive amplitude scale $\Omega$ in this convention.

Why Fast Terms Average Out

A rapidly oscillating contribution has integrals of the form

$$\int_0^T dt\,B(t)e^{i\Lambda t}, \qquad \lvert\Lambda\rvert\gg \frac1T,$$

where $B(t)$ varies slowly compared with $e^{i\Lambda t}$. Repeated integration by parts or stationary-phase reasoning shows that the leading contribution is suppressed by powers of $1/\Lambda$.

For the driven two-level system, the fast frequency is roughly

$$\Lambda\approx\omega_0+\omega.$$

Thus the counter-rotating contribution is small when

$$\frac{\Omega}{\omega_0+\omega}\ll1.$$

This is an averaging argument, not a symmetry argument. The dropped terms are not forbidden; they are perturbatively small under specified conditions.

Validity Conditions

The rotating-wave approximation is typically controlled when the following conditions hold:

• The drive amplitude is weak compared with the fast frequency scale.
• The detuning is small compared with the transition frequency but not so small that neglected shifts matter.
• The drive envelope changes slowly over an optical, microwave, or Larmor cycle.
• The observation time is not so long that small counter-rotating effects accumulate beyond the desired accuracy.
• Other transitions are sufficiently far away or forbidden by selection rules.

A compact diagnostic is

$$\lvert\Omega\rvert, \lvert\Delta\rvert \ll \omega_0+\omega.$$

The relevant dimensionless error estimate depends on the observable. For spectroscopy, even a small frequency shift can matter.

Bloch-Siegert Shift

The counter-rotating terms do not merely disappear. At next order, they shift the resonance frequency. This correction is called the Bloch-Siegert shift.

For the simple driven two-level model, its scale is

$$\delta\omega_{\mathrm{BS}} \sim \frac{\Omega^2}{\omega_0+\omega}.$$

The numerical coefficient depends on the precise convention used for $\Omega$ and on the drive Hamiltonian. The important point is the scaling: the shift is second order in drive amplitude and suppressed by the fast frequency.

In weak-drive spectroscopy this shift may be negligible. In strong driving, ultrastrong light-matter coupling, or precision control, it must be included.

Relation to Harmonic Perturbations

In first-order transition theory, harmonic perturbations produce terms with phase factors

$$e^{i(\omega_{fi}-\omega)t} \quad\text{and}\quad e^{i(\omega_{fi}+\omega)t}.$$

The rotating-wave approximation keeps the phase-matched term and discards the term whose phase rotates rapidly. Thus the RWA can be viewed as a controlled refinement of the resonance logic used in transition-rate calculations.

The same idea extends beyond two levels. In a multilevel system, each term must be judged in the rotating frame appropriate to its transition frequency. A term that is fast for one transition may be resonant for another.

Relation to Effective Hamiltonians

The RWA is an effective-Hamiltonian construction in time rather than in energy-subspace projection. A fast scale is removed, and its leading effect is encoded in a simpler generator of slow dynamics.

Compared with the Schrieffer–Wolff transformation, the RWA is less about eliminating a high-energy subspace and more about eliminating rapidly oscillating Fourier components. Both require a small ratio and both can generate higher-order shifts when pushed beyond leading order.

Adiabatic Elimination removes a fast amplitude after the rotating frame has been chosen. The RWA may first make the couplings slowly varying, but it does not by itself justify removing an excited state or cavity mode.

The Magnus expansion gives a systematic language for this averaging viewpoint: a periodically driven Hamiltonian can be replaced, over one period, by an effective exponential generator plus corrections involving commutators of the time-dependent Hamiltonian.

Common Mistakes

• Dropping counter-rotating terms before choosing the relevant rotating or interaction frame.
• Applying the RWA far from resonance, where the supposed slow term is not actually slow.
• Assuming the Bloch-Siegert shift is always negligible.
• Forgetting other nearby transitions in a multilevel system.
• Treating $\Omega$ as a universal Rabi frequency without checking the convention in the Hamiltonian.
• Using the RWA for very short pulses whose bandwidth is comparable to the fast scale.

Cross-Links

• Effective Hamiltonians and Scale Separation
• Harmonic Perturbations
• Resonant Driving
• Rabi Formula in the Weak-Drive Limit
• Rabi Oscillations: First Encounter
• Pulse-Level Control
• Selection Rules and Transition Rates
• Interaction Picture
• Adiabatic Elimination
• Magnus Expansion
• Schrieffer–Wolff Transformation
• Two-Level System
• Jaynes–Cummings Model

References

• I. I. Rabi, “Space quantization in a gyrating magnetic field,” Physical Review 51, 652-654, 1937.
• F. Bloch and A. Siegert, “Magnetic resonance for nonrotating fields,” Physical Review 57, 522-527, 1940.
• C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Wiley, 1992.
• B. W. Shore, The Theory of Coherent Atomic Excitation, Wiley, 1990.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Starting from

$$V_I(t) = \frac{\hbar\Omega}{2} \big[ \sigma_+e^{i(\omega_0+\omega)t} +\sigma_+e^{i(\omega_0-\omega)t} +\sigma_-e^{-i(\omega_0-\omega)t} +\sigma_-e^{-i(\omega_0+\omega)t} \big],$$

identify the retained terms when $\omega\approx\omega_0$.

Solution

The detuning

$$\Delta=\omega_0-\omega$$

is small. The slowly rotating terms are

$$\sigma_+e^{i\Delta t}, \qquad \sigma_-e^{-i\Delta t}.$$

The terms with $\omega_0+\omega$ rotate rapidly and are dropped at leading RWA order.

2. A driven two-level system has $\Omega/(\omega_0+\omega)=10^{-3}$. Should the RWA be trusted for a rough transition-probability estimate? Should the same answer automatically hold for a precision frequency measurement?

Solution

For a rough transition-probability estimate, the small ratio suggests that counter-rotating amplitudes are strongly suppressed, so the RWA is likely adequate if no other nearby transitions exist.

For a precision frequency measurement, the answer is not automatic. The counter-rotating terms can shift the resonance by a scale of order

$$\frac{\Omega^2}{\omega_0+\omega}.$$

Even if this is tiny as a dimensionless fraction, it may be experimentally resolvable.